Theorem fvprif 13506

Description: The value of the pair function at an element of 2o. (Contributed by Mario Carneiro, 14-Aug-2015.)

Assertion

Ref	Expression
fvprif	|- ( ( A e. V /\ B e. W /\ C e. 2o ) -> ( { <. (/) , A >. , <. 1o , B >. } ` C ) = if ( C = (/) , A , B ) )

Proof of Theorem fvprif

Step	Hyp	Ref	Expression
1		fvpr0o 13504	. . . . 5 |- ( A e. V -> ( { <. (/) , A >. , <. 1o , B >. } ` (/) ) = A )
2	1	3ad2ant1 1045	. . . 4 |- ( ( A e. V /\ B e. W /\ C e. 2o ) -> ( { <. (/) , A >. , <. 1o , B >. } ` (/) ) = A )
3	2	adantr 276	. . 3 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = (/) ) -> ( { <. (/) , A >. , <. 1o , B >. } ` (/) ) = A )
4		simpr 110	. . . 4 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = (/) ) -> C = (/) )
5	4	fveq2d 5652	. . 3 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = (/) ) -> ( { <. (/) , A >. , <. 1o , B >. } ` C ) = ( { <. (/) , A >. , <. 1o , B >. } ` (/) ) )
6	4	iftrued 3616	. . 3 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = (/) ) -> if ( C = (/) , A , B ) = A )
7	3, 5, 6	3eqtr4d 2274	. 2 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = (/) ) -> ( { <. (/) , A >. , <. 1o , B >. } ` C ) = if ( C = (/) , A , B ) )
8		fvpr1o 13505	. . . . 5 |- ( B e. W -> ( { <. (/) , A >. , <. 1o , B >. } ` 1o ) = B )
9	8	3ad2ant2 1046	. . . 4 |- ( ( A e. V /\ B e. W /\ C e. 2o ) -> ( { <. (/) , A >. , <. 1o , B >. } ` 1o ) = B )
10	9	adantr 276	. . 3 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = 1o ) -> ( { <. (/) , A >. , <. 1o , B >. } ` 1o ) = B )
11		simpr 110	. . . 4 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = 1o ) -> C = 1o )
12	11	fveq2d 5652	. . 3 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = 1o ) -> ( { <. (/) , A >. , <. 1o , B >. } ` C ) = ( { <. (/) , A >. , <. 1o , B >. } ` 1o ) )
13		1n0 6643	. . . . . 6 |- 1o =/= (/)
14	13	neii 2405	. . . . 5 |- -. 1o = (/)
15	11	eqeq1d 2240	. . . . 5 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = 1o ) -> ( C = (/) <-> 1o = (/) ) )
16	14, 15	mtbiri 682	. . . 4 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = 1o ) -> -. C = (/) )
17	16	iffalsed 3619	. . 3 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = 1o ) -> if ( C = (/) , A , B ) = B )
18	10, 12, 17	3eqtr4d 2274	. 2 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = 1o ) -> ( { <. (/) , A >. , <. 1o , B >. } ` C ) = if ( C = (/) , A , B ) )
19		elpri 3696	. . . 4 |- ( C e. { (/) , 1o } -> ( C = (/) \/ C = 1o ) )
20		df2o3 6640	. . . 4 |- 2o = { (/) , 1o }
21	19, 20	eleq2s 2326	. . 3 |- ( C e. 2o -> ( C = (/) \/ C = 1o ) )
22	21	3ad2ant3 1047	. 2 |- ( ( A e. V /\ B e. W /\ C e. 2o ) -> ( C = (/) \/ C = 1o ) )
23	7, 18, 22	mpjaodan 806	1 |- ( ( A e. V /\ B e. W /\ C e. 2o ) -> ( { <. (/) , A >. , <. 1o , B >. } ` C ) = if ( C = (/) , A , B ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 716   /\ w3a 1005   = wceq 1398   e. wcel 2202   (/)c0 3496   ifcif 3607   {cpr 3674   <.cop 3676   ` cfv 5333   1oc1o 6618   2oc2o 6619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-id 4396  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-res 4743  df-iota 5293  df-fun 5335  df-fv 5341  df-1o 6625  df-2o 6626

This theorem is referenced by: (None)